449 research outputs found
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
Edge-Stable Equimatchable Graphs
A graph is \emph{equimatchable} if every maximal matching of has the
same cardinality. We are interested in equimatchable graphs such that the
removal of any edge from the graph preserves the equimatchability. We call an
equimatchable graph \emph{edge-stable} if , that is the
graph obtained by the removal of edge from , is also equimatchable for
any . After noticing that edge-stable equimatchable graphs are
either 2-connected factor-critical or bipartite, we characterize edge-stable
equimatchable graphs. This characterization yields an time recognition algorithm. Lastly, we introduce and shortly
discuss the related notions of edge-critical, vertex-stable and vertex-critical
equimatchable graphs. In particular, we emphasize the links between our work
and the well-studied notion of shedding vertices, and point out some open
questions
Matchability of heterogeneous networks pairs
We consider the problem of graph matchability in non-identically distributed networks. In a general class of edge-independent networks, we demonstrate that graph matchability is almost surely lost when matching the networks directly, and is almost perfectly recovered when first centering the networks using Universal Singular Value Thresholding before matching. These theoretical results are then demonstrated in both real and synthetic simulation settings. We also recover analogous core-matchability results in a very general core-junk network model, wherein some vertices do not correspond between the graph pair.First author draf
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